## Geometry, Old and New, Its Problems and Principles: A Paper |

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Geometry, Old and New, Its Problems and Principles: A Paper Benjamin Gratz Brown No preview available - 2009 |

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ancients application arcs arithmetical assume base beauty becomes bounded called century circle circumference close conception condition consider construction contained course curve definite demonstration describe designation determination diagram diameter direction divided drawn elements entirely equal equation establishes exist expression extension extremity fact figures fixed former four geometry give given greater half human idea important indicated infinite inscribed interest intersection known less likewise limit magnitudes manner mathematics matter means measure method mind multiple nature numbers obtained opposite origin parallel parallelogram perhaps perpendicular Persian plane position present problems properties proportion propositions pure radius ratio reasoning reference regard relations remains represent result right angles right triangle seen segment shown sides similar simple solution space square straight line studies surface taken thing thought tion triangle true truth unit volume whilst whole wonder

### Popular passages

Page 14 - In any triangle, the product of two sides is equal to the product of the segments of the third side formed by the bisector of the opposite angle plus the square of the bisector.

Page 19 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.

Page 10 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.

Page 17 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sidef.

Page 23 - ... distance between the centres of the inscribed circle and of the circle through the middle points of sides has been proved to be exactly the difference between their radii ; and the same argument applies to any of the four circles which touch the three sides of the given triangle ; hence (5) The circle which passes through the middle points of the sides of a triangle touches the four circles which touch the three sides. This theorem was new both to Dr Hart and myself,* but I have lately learned...

Page 8 - THEOREM. 35. Of two oblique lines drawn from the same point to the same straight line, that is the greater which cuts off upon the line the greater distance from the perpendicular. Let PC be the perpendicular from P to AB, and suppose CE > CD; thenPU> PD. For, produce PC to P', making CP

Page 15 - Two triangles are similar if an angle of one equals an angle of the other and the sides including these angles are proportional.

Page 18 - Two diagonals of a regular pentagon, not drawn from a common vertex, divide each other in extreme and mean ratio.

Page 31 - The surface of a sphere is equal to the product of its diameter by the circumference of a great circle.

Page 16 - In any triangle, if a straight line is drawn from the vertex to the middle of the base, the sum of the squares of the other two sides is equivalent to twice the square of the bisecting line, together with twice the square of half the base.